Skip to main content
SpringerLink
Log in

Lectures on the Topological Vertex

  • Chapter
Enumerative Invariants in Algebraic Geometry and String Theory

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1947))

In this lectures, I will summarize the approach to Gromov–Witten invariants on toric Calabi–Yau threefolds based on large N dualities. Since the large N duality/topological vertex approach computes Gromov–Witten invariants in terms of Chern–Simons knot and link invariants, Sect. 2 is devoted to a review of these. Section 3 reviews topological strings and Gromov–Witten invariants, and gives some information about the open string case. Section 4 introduces the class of geometries we will deal with, namely toric (noncompact) Calabi–Yau manifolds, and we present a useful graphical way to represent these manifolds which constitutes the geometric core of the theory of the topological vertex. Finally, in Sect. 5, we define the vertex and present some explicit formulae for it and some simple applications. A brief Appendix contains useful information about symmetric polynomials.

It has not been possible to present all the relevant background and physical derivations in this set of lectures. However, these topics have been extensively reviewed for example in the book Mariño (2005), to which we refer for further information and/or references.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aganagic, M., Klemm, A., Mariño, M., and Vafa, C. (2005). The topological vertex. Commun. Math. Phys. 254, 425–478; eprint hep-th/0305132.

    Article  MATH  Google Scholar 

  2. Aganagic, M., Klemm, A., and Vafa, C. (2002). Disk instantons, mirror symmetry and the duality web. Z. Naturforsch. A 57, 1–28; eprint hep-th/0105045.

    MathSciNet  Google Scholar 

  3. Aganagic, M., Mariño, M., and Vafa, C. (2004). All loop topological string amplitudes from Chern–Simons theory. Commun. Math. Phys. 247, 467–512; eprint hep-th/0206164.

    Article  MATH  Google Scholar 

  4. Aganagic, M. and Vafa, C. (2000). Mirror symmetry, D-branes and counting holomorphic discs. Eprint hep-th/0012041.

    Google Scholar 

  5. Aspinwall, P. S., and Morrison, D. R. (1993). Topological field theory and rational curves. Commun. Math. Phys. 151, 245–62; eprint hep-th/9110048.

    Article  MATH  MathSciNet  Google Scholar 

  6. Atiyah, M. (1990). On framings of three-manifolds. Topology 29, 1–7.

    Article  MATH  MathSciNet  Google Scholar 

  7. Axelrod, S. and Singer, I. (1992). Chern–Simons perturbation theory. In Catto, Sultan (ed.) et al., Differential geometric methods in theoretical physics, World Scientific, Singapore, pp. 3–45; eprint hep-th/9110056.

    Google Scholar 

  8. Bar-Natan, D. (1995). On the Vassiliev knot invariants. Topology 34, 423–72.

    Article  MATH  MathSciNet  Google Scholar 

  9. Bershadsky, M., Cecotti, S., Ooguri, H., and Vafa, C. (1993). Holomorphic anomalies in topological field theories. Nucl. Phys. B 405, 279–304; eprint hep-th/9302103.

    Article  MathSciNet  Google Scholar 

  10. Bershadsky, M., Cecotti, S., Ooguri, H., and Vafa, C. (1994). Kodaira–Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165, 311–428; eprint hep-th/9309140.

    Article  MATH  MathSciNet  Google Scholar 

  11. Bryan, J. and Karp, D. (2005). The closed topological vertex via the Cremona transform. J. Algebraic Geom. 14, 529–42; eprint math.AG/0311208.

    MATH  MathSciNet  Google Scholar 

  12. Bryan, J. and Pandharipande, R. (2001). BPS states of curves in Calabi–Yau 3-folds. Geom. Topol. 5, 287–318; eprint math.AG/0009025.

    Article  MATH  MathSciNet  Google Scholar 

  13. Camperi, M., Levstein, F., and Zemba, G. (1990). The large N limit of Chern–Simons theory. Phys. Lett. B 247, 549–54.

    Article  MathSciNet  Google Scholar 

  14. Candelas, P., De La Ossa, X. C., Green, P., and Parkes, L. (1991). A pair of Calabi–Yau manifolds as an exactly soluble superconformal theory. Nucl. Phys. B 359, 21–74.

    Article  MATH  Google Scholar 

  15. Chiang, T. M., Klemm, A., Yau, S. T., and Zaslow, E. (1999). Local mirror symmetry: Calculations and interpretations. Adv. Theor. Math. Phys. 3, 495–565; eprint hep-th/9903053.

    MATH  MathSciNet  Google Scholar 

  16. Correale, R. and Guadagnini, E. (1994). Large N Chern–Simons field theory. Phys. Lett. B 337, 80–5.

    Article  MathSciNet  Google Scholar 

  17. Cox, D. and Katz, S. 1999. Mirror symmetry and algebraic geometry, American Mathematical Society, Providence.

    Google Scholar 

  18. Cvitanovic, P. (1976). Group theory for Feynman diagrams in nonabelian gauge theories. Phys. Rev. D 14, 1536–53.

    Article  MathSciNet  Google Scholar 

  19. Cvitanovic, P. (2004). Group theory. Birdtracks, Lie’s, and exceptional groups. In http://www.cns.gatech.edu/GroupTheory.

  20. Diaconescu, D. E. and Florea, B. (2005). Localization and gluing of topological amplitudes. Commun. Math. Phys. 257, 119–49; eprint hep-th/0309143.

    Article  MATH  MathSciNet  Google Scholar 

  21. Di Francesco, P., Mathieu, P., and Sénéchal, D. (1997). Conformal field theory, Springer-Verlag, New York.

    MATH  Google Scholar 

  22. Dijkgraaf, R. (1995). Perturbative topological field theory. In String theory, gauge theory and quantum gravity ’93, 189–227, World Scientific.

    Google Scholar 

  23. Elitzur, S., Moore, G., Schwimmer, A., and Seiberg, N. (1989). Remarks on the canonical quantization of the Chern–Simons-Witten theory. Nucl. Phys. B 326, 108–34.

    Article  MathSciNet  Google Scholar 

  24. Faber, C. and Pandharipande, R. (2000). Hodge integrals and Gromov–Witten theory. Invent. Math. 139, 173–99; eprint math.AG/9810173.

    Article  MATH  MathSciNet  Google Scholar 

  25. Freed, D. S. and Gompf, R. E. (1991). Computer calculation of Witten’s three-manifold invariant. Commun. Math. Phys. 141, 79–117.

    Article  MATH  MathSciNet  Google Scholar 

  26. Freyd, P., Yetter, D., Hoste, J., Lickorish, W.B.R., Millett, K., and Ocneanu, A.A. (1985). A new polynomial invariant of knots and links. Bull. Am. Math. Soc. (N.S.) 12, 239–46.

    Article  MATH  MathSciNet  Google Scholar 

  27. Fulton, W. and Harris, J. (1991). Representation theory. A first course. Springer-Verlag, New York.

    MATH  Google Scholar 

  28. Gopakumar, R., and Vafa, C. (1998a). M-theory and topological strings, I. Eprint hep-th/9809187.

    Google Scholar 

  29. Gopakumar, R. and Vafa, C. (1998b). M-theory and topological strings, II. Eprint hep-th/9812127.

    Google Scholar 

  30. Gopakumar, R. and Vafa, C. (1999). On the gauge theory/geometry correspondence. Adv. Theor. Math. Phys. 3, 1415–43; eprint hep-th/9811131.

    MATH  MathSciNet  Google Scholar 

  31. Graber, T. and Zaslow, E. (2002). Open-string Gromov-Witten invariants: Calculations and a mirror “theorem”. Contemp. Math. 310, 107–121; eprint hep-th/0109075.

    MathSciNet  Google Scholar 

  32. Griffiths, P. and Harris, J. (1977). Principles of algebraic geometry, John Wiley and Sons, New York.

    Google Scholar 

  33. Guadagnini, E., Martellini, M., and Mintchev, M. (1990). Wilson lines in Chern–Simons theory and link invariants. Nucl. Phys. B 330, 575–607.

    Article  MathSciNet  Google Scholar 

  34. Harris, J. and Morrison, I. (1998). Moduli of curves. Springer-Verlag, New York.

    MATH  Google Scholar 

  35. Harvey, R. and Lawson, H.B. (1982). Calibrated geometries. Acta Math. 148, 47–157.

    Article  MATH  MathSciNet  Google Scholar 

  36. Hori, K., Katz, S., Klemm, A., Pandharipande, R., Thomas, R., Vafa, C., Vakil, R., and Zaslow, E. (2003). Mirror symmetry. American Mathematical Society, Providence.

    Google Scholar 

  37. Iqbal, A. (2002). All genus topological string amplitudes and 5-brane webs as Feynman diagrams. Eprint hep-th/0207114.

    Google Scholar 

  38. Jeffrey, L. C. (1992). Chern–Simons-Witten invariants of lens spaces and torus bundles, and the semi-classical approximation. Commun. Math. Phys. 147, 563–604.

    Article  MATH  MathSciNet  Google Scholar 

  39. Karp, D., Liu, C.-C. M., and Mariño, M. (2005). The local Gromov–Witten invariants of configurations of rational curves. Eprint math.AG/0506488.

    Google Scholar 

  40. Katz, S., Klemm, A., and Vafa, C. (1999). M-theory, topological strings and spinning black holes. Adv. Theor. Math. Phys. 3, 1445–537; eprint hep-th/9910181.

    MATH  MathSciNet  Google Scholar 

  41. Katz, S., and Liu, C.-C. (2002). Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc. Adv. Theor. Math. Phys. 5, 1–49; eprint math.ag/0103074.

    MathSciNet  Google Scholar 

  42. Klemm, A. and Zaslow, E. (2001). Local mirror symmetry at higher genus. In Winter School on Mirror Symmetry, Vector bundles and Lagrangian Submanifolds, American Mathematical Society, Providence, p. 183–207; eprint hep-th/9906046.

    Google Scholar 

  43. Kontsevich, M. (1995). Enumeration of rational curves via torus actions. Prog. Math. 129, 335–68; eprint hep-th/9405035.

    MathSciNet  Google Scholar 

  44. Labastida, J.M.F. (1999). Chern–Simons theory: ten years after. In Trends in Theoretical Physics II, H. Falomir, R. E. Gamboa and F. A. Schaposnik, eds., AIP Conference Proceedings, Vol. 484, p. 1–40; eprint hep-th/9905057.

    Google Scholar 

  45. Labastida, J. M. F. and Ramallo, A. V. (1989). Operator formalism for Chern–Simons theories. Phys. Lett. B 227, 92–102.

    Article  MathSciNet  Google Scholar 

  46. Le, T. T. Q., Murakami, J., and Ohtsuki, T. (1998). On a universal perturbative invariant of 3-manifolds. Topology 37, 539–74.

    Article  MATH  MathSciNet  Google Scholar 

  47. Li, J., Liu, C.-C. M., Liu, K., and Zhou, J. (2004). A mathematical theory of the topological vertex. Eprint math.AG/0408426.

    Google Scholar 

  48. Li, J. and Song, Y. S. (2002). Open string instantons and relative stable morphisms. Adv. Theor. Math. Phys. 5, 67–91; eprint hep-th/0103100.

    MathSciNet  Google Scholar 

  49. Lickorish, W. B. R. (1998). An introduction to knot theory, Springer-Verlag, New York.

    Google Scholar 

  50. Macdonald, I.G. (1995). Symmetric functions and Hall polynomials, 2nd edition, Oxford University Press, Oxford.

    MATH  Google Scholar 

  51. Mariño, M. (2005). Chern–Simons theory, matrix models, and topological strings. Oxford University Press.

    Google Scholar 

  52. Mariño, M. and Vafa, C. (2002). Framed knots at large N. Contemp. Math. 310, 185–204; eprint hep-th/0108064.

    Google Scholar 

  53. Maulik, D., Nekrasov, N., Okounkov, A., and Pandharipande, R. (2003). Gromov–Witten theory and Donaldson–Thomas theory. Eprint math.AG/ 0312059.

    Google Scholar 

  54. Mayr, P. (2002). Summing up open string instantons and \({\cal N} = 1\) string amplitudes. Eprint hep-th/0203237.

    Google Scholar 

  55. Morton, H.R. and Lukac, S. G. (2003). The HOMFLY polynomial of the decorated Hopf link. J. Knot Theory Ramifications 12, 395–416; eprint math.GT/0108011.

    Article  MATH  MathSciNet  Google Scholar 

  56. Ohtsuki, T. (2002). Quantum invariants. World Scientific, Singapore.

    MATH  Google Scholar 

  57. Okounkov, A., Reshetikhin, N., and Vafa, C. (2003). Quantum Calabi–Yau and classical crystals. Eprint hep-th/0309208.

    Google Scholar 

  58. Ooguri, H. and Vafa, C. (2000). Knot invariants and topological strings. Nucl. Phys. B 577, 419–38; eprint hep-th/9912123.

    Article  MATH  MathSciNet  Google Scholar 

  59. Ooguri, H. and Vafa, C. (2002). Worldsheet derivation of a large N duality. Nucl. Phys. B 641, 3–34; eprint hep-th/0205297.

    Google Scholar 

  60. Periwal, V. (1993). Topological closed string interpretation of Chern–Simons theory. Phys. Rev. Lett. 71, 1295–8; eprint hep-th/9305115.

    Article  MATH  MathSciNet  Google Scholar 

  61. Polyakov, A. M. (1988). Fermi-Bose transmutations induced by gauge fields. Mod. Phys. Lett. A 3, 325–8.

    Article  MathSciNet  Google Scholar 

  62. Prasolov, V. V. and Sossinsky, A. B. (1997). Knots, links, braids and 3-manifolds, American Mathematical Society, Providence.

    Google Scholar 

  63. Rozansky, L. (1995). A large k asymptotics of Witten’s invariant of Seifert manifolds. Commun. Math. Phys. 171, 279–322; eprint hep-th/9303099.

    Article  MATH  MathSciNet  Google Scholar 

  64. Rozansky, L. and Witten, E. (1997). Hyper-Kähler geometry and invariants of three-manifolds. Selecta Math. 3, 401–58; eprint hep-th/9612216.

    Article  MATH  MathSciNet  Google Scholar 

  65. Sawon, J. (2004). Rozansky-Witten invariants of hyperKähler manifolds. Eprint math.DG/0404360.

    Google Scholar 

  66. Schwarz, A. (1987). New topological invariants arising in the theory of quantized fields. Baku International Topological Conference, Abstracts (Part 2) Baku.

    Google Scholar 

  67. ’t Hooft, G. (1974). A planar diagram theory for strong interactions. Nucl. Phys. B 72 461–73.

    Article  MathSciNet  Google Scholar 

  68. Witten, E. (1989). Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351–99.

    Article  MATH  MathSciNet  Google Scholar 

  69. Witten, E. (1993). Phases of \({\cal N} = 2\) theories in two dimensions. Nucl. Phys. B 403, 159-222; eprint hep-th/9301042.

    Google Scholar 

  70. Witten, E. (1995). Chern–Simons gauge theory as a string theory. Prog. Math. 133, 637–78; eprint hep-th/9207094.

    MathSciNet  Google Scholar 

  71. Zhou, J. (2003). Localizations on moduli spaces and free field realizations of Feynman rules. Eprint math.AG/0310283.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Theory Division, CERN, University of Geneva, 1211 Geneva, Switzerland

    M. Mariño

Authors
  1. M. Mariño
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of British Columbia, 1984 Mathematics Rd, Vancouver, V6T 1Z2, BC, Canada

    Kai Behrend

  2. Department of Mathematics “Guido Castelnuovo”, University of Rome “La Sapienza”, P.le Aldo Moro 5, 00185 Rome, Italy

    Marco Manetti

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Mariño, M. (2008). Lectures on the Topological Vertex. In: Behrend, K., Manetti, M. (eds) Enumerative Invariants in Algebraic Geometry and String Theory. Lecture Notes in Mathematics, vol 1947. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79814-9_2

Download citation

Publish with us

Policies and ethics

Search

Navigation